Cars arrive randomly at a tollbooth at a rate of 15 cars per 10 minutes during rush hour. What is the probability that exactly five cars will arrive over a five-minute interval during rush hour?Multiple Choice0.12160.10940.57160.1967

The problem can be solved using the Poisson distribution formula. The Poisson distribution is a probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event.

The formula for the Poisson distribution is:

P(x; μ) = (e^-μ) * (μ^x) / x!

where:

• P(x; μ) is the probability of x events in an interval,
• e is the base of the natural logarithm (approximately equal to 2.71828),
• μ is the average rate of value (mean) for the given interval,
• x is the actual number of successes that result from the experiment.

Given that the rate of cars is 15 cars per 10 minutes, we can calculate the rate for 5 minutes, which is half of the 10-minute rate. So, μ = 15 cars / 2 = 7.5 cars.

We want to find the probability that exactly 5 cars will arrive over a 5-minute interval. So, x = 5.

Substituting these values into the formula, we get:

P(5; 7.5) = (e^-7.5) * (7.5^5) / 5!

Calculating this gives us a probability of approximately 0.1094.

So, the correct answer is 0.1094.